This paper addresses the multi-rate stabilization problem for linear singularly perturbed systems. The proposed multi-rate sampled-data control law is based on the discretization in multi-rate fashion on the continuous-time composite control law obtained from the singular perturbation theory. The sampling times of the slow and fast state variables are allowed to be asynchronous and nonuniformly spaced. A new time-dependent Lyapunov functional is introduced to analyze the closed-loop stability of the considered system with the multi-rate feedback. With the use of the Lyapunov functional, a sufficient condition for exponential stability of the closed-loop system is derived in terms of linear matrix inequalities. Further, a robust stabilizability condition of the proposed multi-rate control law with respect to uncertain singular perturbation parameter is also obtained. Three numerical examples are presented to show the effectiveness of the developed methodology.

本文讨论了线性奇异摄动系统的多速率稳定问题。所提出的多速率采样数据控制律是基于从奇异摄动理论获得的连续时间复合控制律的多速率离散化。慢速状态变量和快速状态变量的采样时间允许异步且间隔不均匀。引入了一种新的基于时间的Lyapunov函数，以分析具有多速率反馈的所考虑系统的闭环稳定性。使用Lyapunov函数，就线性矩阵不等式得出了闭环系统指数稳定性的充分条件。进一步，还获得了针对不确定奇异摄动参数的拟定多速率控制律的鲁棒镇定条件。给出了三个数值示例，以说明所开发方法的有效性。